Quadratic formula

The quadratic formula is the formula for finding the solutions of quadratic equations. Students are sometimes taught a method known as factoring, but that's really just looking for a lucky guess.

First, the quadratic equation must be reduced to this format:

Then the coefficients a, b, and c can be substituted in the formula to find the solutions:

The formula is derived in the following way, which is known as completing the square:

Now we need to get something of the form that matches the first two terms. We have

So we need to get a match.

Plugging that in, we get

You can check that the formula is correct by substituting the formula (with either sign for the square root) in place of x in and then gradually simplifying the rather complicated formula that results, step by step. Eventually, if all the steps are done correctly, it will simplify to 0.